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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Review of Shadows of the Mind
Message-ID: <1995Mar27.142154.3381@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Mon, 27 Mar 1995 14:21:54 GMT
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ghrosenb@phil.indiana.edu (Gregg Rosenberg) writes:

> arromdee@jyusenkyou.cs.jhu.edu (Ken Arromdee) writes:

>>What about the statement "this statement is true, but Roger Penrose
>>cannot consistently accept it"?  If Penrose believes this statement
>>false, he has made an error, but he cannot in principle be shown that
>>it is an error.

>That is not the form of Penrose's argument. His argument relies on the
>fact that, given a formal system K, we can see that its Godel formula
>follows from its consistency via the normal Godel reasoning. This is a
>rigorously proven, mathematical theorem, that applies to *any* formal
>system whatsoever.

And, as Ken points out, it holds to Roger Penrose, as well. The "Godel
formula" for Penrose is:

    G(Penrose) =
    "This statement is not considered an unassailable truth by
     Roger Penrose."

Now, it is obviously inconsistent for Penrose to accept G(Penrose) as
an unassailable truth. Therefore, if Penrose' unassailable truths are
consistent, then G(Penrose) is not one of them. But since G(Penrose)
holds if and only if G(Penrose) is not an unassailable truth of Roger
Penrose, it follows that if Penrose' unassailable truths are
consistent, then G(Penrose) is true. Therefore, if Penrose *believes*
(unassailably) that his unassailable truths were consistent, then he
would be forced, by modus-ponens, to believe G(Penrose). But it is
inconsistent for Penrose to believe G(Penrose).

Conclusion: If Roger Penrose is consistent, then the fact that he is
consistent is not one of his unassailable beliefs.

Penrose' mistake is in thinking that Godel's incompleteness theorem
only applies to computable theories. It applies to a much larger
collection of theories than that. Basically, no consistent theory
(whether or computable or not) can have as a consequence that it is
itself consistent.

>There are no funny self-referential tricks which stop us from being
>able to derive the conditional: if K is consistent, then G(K) is
>true. Now, assume we are K. We know we are (in principle) consistent,
>so by modus ponens we know G(K) is true.  Yet K could not know G(K) is
>true, on pain of contradiction. So we must conclude that we are not K
>(or accept that 1=2, since we would be accepting that we are
>inconsistent). This is true for any K whatsoever, so we must not be
>any K.

The problem is that we *don't* know that we are in principle
consistent. Actually, some of us *think* they know but they are
wrong---those people who think they are consistent actually aren't.

Daryl McCullough
ORA Corp.
Ithaca, NY
